Estimation of risk measures for extreme pluviometrical measurements

Jonathan El Methni (1) , Laurent Gardes (2) , Stéphane Girard (3)

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
2 IRMA - Institut de Recherche Mathématique Avancée
3 MISTIS - Modelling and Inference of Complex and Structured Stochastic Systems

Extreme rainfall statistics are often used when a flood has occurred to assess the rarity of such an event. A typical problem is to estimate the amount that will fall on a day of exceptionally heavy rainfall which is expected to occur every T years. Usually, hydrologists are interested in the value T = 100 corresponding to a centenary event. Statistically speaking, the problem is to estimate the T-year return level which is a upper quantile of the distribution of the variable of interest also called Value-at-Risk. This risk measure however suffers from several weaknesses. The estimation of a single extreme quantile only gives incomplete information on the extremes of a random variable. To put it differently, it may well be the case that a light-tailed distribution (e.g. a Gaussian distribution) and a heavy-tailed distribution share a quantile at some common level, although they clearly do not have the same behavior in their extremes. An alternative risk measure is the Conditional Tail Expectation which is the mean of the rainfalls larger than the Value-at-Risk. This risk measure thus takes into account the whole information contained in the upper tail of the distribution. It is frequently encountered in financial investment or in the insurance industry Here, we focus on the estimation of these risk measures in case of extreme rainfall modeled by heavy-tailed distributions. In order to take into account the geographical factors, we also assume that these risk measures depend on a covariate. We present the theoretical properties of our esti-mators. The behaviour and the efficiency of our estimators are illustrated on rainfall observations in the Cévennes-Vivarais region (southern part of France). This data set is provided by the French meteorological service Météo-France and consists in daily rainfalls measured at 523 raingauge stations from 1958 to 2000. In this context, the variable of interest is the daily rainfall measured in millimeters. The covariate is the three dimensional geographical location (longitude, latitude and altitude).

Mots clés: risk measure; heavy-tailed distributions; kernel estimators; extreme-value statistics.

Domaines: Statistiques [math.ST]

Jonathan El Methni, Laurent Gardes, Stéphane Girard. Estimation of risk measures for extreme pluviometrical measurements. 26th Annual Conference of The International Environmetrics Society, Jul 2016, Edimbourg, United Kingdom. ⟨hal-01350104⟩